## Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators

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- by Steve Hofmann, Carlos Kenig, Svitlana Mayboroda and Jill Pipher
- J. Amer. Math. Soc.
**28**(2015), 483-529 - DOI: https://doi.org/10.1090/S0894-0347-2014-00805-5
- Published electronically: May 21, 2014
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## Abstract:

We consider divergence form elliptic operators $L= {-}\mathrm {div} A(x) \nabla$, defined in the half space $\mathbb {R}^{n+1}_+$, $n\geq 2$, where the coefficient matrix $A(x)$ is bounded, measurable, uniformly elliptic, $t$-independent, and not necessarily symmetric. We establish square function/non-tangential maximal function estimates for solutions of the homogeneous equation $Lu=0$, and we then combine these estimates with the method of “$\epsilon$-approximability” to show that $L$-harmonic measure is absolutely continuous with respect to surface measure (i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant sense: more precisely, that it belongs to the class $A_\infty$ with respect to surface measure (equivalently, that the Dirichlet problem is solvable with data in $L^p$, for some $p<\infty$). Previously, these results had been known only in the case $n=1$.## References

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## Bibliographic Information

**Steve Hofmann**- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 251819
- ORCID: 0000-0003-1110-6970
- Email: hofmanns@missouri.edu
**Carlos Kenig**- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois, 60637
- MR Author ID: 100230
- Email: cek@math.chicago.edu
**Svitlana Mayboroda**- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455
- MR Author ID: 739839
- Email: svitlana@math.umn.edu
**Jill Pipher**- Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
- MR Author ID: 237541
- Email: jpipher@math.brown.edu
- Received by editor(s): February 10, 2012
- Received by editor(s) in revised form: February 11, 2014
- Published electronically: May 21, 2014
- Additional Notes: Each of the authors was supported by the NSF

This work has been possible thanks to the support and hospitality of the*University of Chicago*, the*University of Minnesota*, the*University of Missouri*,*Brown University*, and the*BIRS Centre in Banff*(Canada). The authors would like to express their gratitude to these institutions. - © Copyright 2014 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**28**(2015), 483-529 - MSC (2010): Primary 42B99, 42B25, 35J25, 42B20
- DOI: https://doi.org/10.1090/S0894-0347-2014-00805-5
- MathSciNet review: 3300700